Problem: $P(x)=x^4-3x^2+kx-2$ where $k$ is an unknown integer. $P(x)$ divided by $(x-2)$ has a remainder of $10$. What is the value of $k$ ? $k=$
Solution: We can use the polynomial remainder theorem to solve this problem: For a polynomial $p(x)$ and a number $a$, the remainder on division by $x-a$ is $p(a)$. According to the theorem, the remainder when $P(x)$ is divided by $(x-{2})$, is equal to $P({2})$. We also know that this remainder is equal to $10$. Therefore, $P({2})=10$. We can use this equality to find $k$. $\begin{aligned} P({2})&=10 \\\\ ({2})^4-3({2})^2+k({2})-2&=10 \\\\ 16-3 \cdot 4+2k-2&=10 \\\\ 2+2k&=10 \\\\ 2k&=8 \\\\ k&=4 \end{aligned}$ In conclusion, $k=4$.